In this article, we analyzed the fifth-order WENO scheme with the
smoothness indicators developed in [1]}, [2]}, [3]}, [4]}, [5]}
and derived a new global smoothness indicator which is a linear combination
of second order derivatives leads to give a fourth-order of accuracy
so, the resulting global smoothness indicator provides a much smoother
information to the evaluation point in evaluating the non-linear weights.
It is verified that the proposed WENO scheme has the fifth-order accuracy
even at critical points where first and second derivatives vanish
but the third derivative is non-zero through the Taylor expansion.
We call this scheme as modified WENO-P (MWENO-P) which takes almost
the same computational cost as that of WENO-NS or WENO-P and is simple
to implement as the WENO-JS or WENO-Z schemes. The numerical experiments
are shown that the proposed scheme MWENO-P performs better than WENO-JS,
WENO-Z, WENO-NS and WENO-P for the problems which contain discontinuities.
